| 1 | = | (1 + p12)−3/2 q1 |, | 1 | = | (1 + p22)−3/2 q2 |. |
| ρ1 | ρ2 |
When p1 and p2 are imaginary the origin is the real point of intersection of two imaginary branches. In the real figure of the curve it is an isolated point. If u2 is a square, a(y − px)2, the origin is a cusp, and in general there is not a series for y in integral powers of x, which is valid in the neighbourhood of the origin. The further investigation of cusps and multiple points belongs rather to analytical geometry and the theory of algebraic functions than to differential calculus.
(x.) When the equation of a curve is given in the form u0 + u1 + ... + un−1 + un = 0 where the notation is the same as that in (ix.), the factors of un determine the directions of the asymptotes. If these factors are all real and distinct, there is an asymptote corresponding to each factor. If un = L1L2 ... Ln, where L1, ... are linear in x, y, we may resolve un−1/un into partial fractions according to the formula
| un−1 | = | A1 | + | A2 | + ... + | An | , |
| un | L1 | L2 | Ln |
and then L1 + A1 = 0, L2 + A2 = 0, ... are the equations of the asymptotes. When a real factor of un is repeated we may have two parallel asymptotes or we may have a “parabolic asymptote.” Sometimes the parallel asymptotes coincide, as in the curve x2(x2 + y2 − a2) = a4, where x = 0 is the only real asymptote. The whole theory of asymptotes belongs properly to analytical geometry and the theory of algebraic functions.
47. The formal definition of an integral, the theorem of the existence of the integral for certain classes of functions, a list of Integral calculus. classes of “integrable” functions, extensions of the notion of integration to functions which become infinite or indeterminate, and to cases in which the limits of integration become infinite, the definitions of multiple integrals, and the possibility of defining functions by means of definite integrals—all these matters have been considered in [Function]. The definition of integration has been explained in § 5 above, and the results of some of the simplest integrations have been given in § 12. A few theorems relating to integrations have been noted in §§ 34, 35, 36 above.
48. Methods of integration.The chief methods for the evaluation of indefinite integrals are the method of integration by parts, and the introduction of new variables.
From the equation d(uv) = u dv + v du we deduce the equation
| ∫ u | dv | dx = uv − ∫ v | du | dx, |
| dx | dx |
or, as it may be written